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# A hollow cylinder, a spherical shell, a solid cylinder and a solid sphere are allowed to roll on a inclined rough surface of coefficient of friction μ and inclination θ. The correct statements are

A
if cylindrical shell can roll on inclined plan, all other objects will also roll
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B
if all the objects are rolling and have same mass, the K.E. of all the objects will be same at the bottom of inclined plane
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C
work done by the frictional force will be zero, if objects are rolling
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D
frictional force will be equal for all the objects, if having same mass
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Solution

## The correct options are A if all the objects are rolling and have same mass, the K.E. of all the objects will be same at the bottom of inclined plane B if cylindrical shell can roll on inclined plan, all other objects will also roll C work done by the frictional force will be zero, if objects are rolling$mg\sin \theta -f=ma$$fr=I\alpha$to satisfy the condition of pure rolling $a=\alpha r$ solving this$f=\frac{mg\sin \theta }{\frac{1}{k^{2}}+1}$since $f=\mu mg\cos \theta$therefore max $\mu =\frac{\tan \theta }{\frac{1}{k^{2}}+1}$thus for cylindrical shell as we know k = 1 which is maximium of all three therefore maximium $\mu$ is offered by the plane and for other two $\mu$ is enough to make them roll.Work done by friction is always 0 in the case of rolling as we know that point of contact is always at rest.Applying the conservation of mechanical energy$\frac{1}{2}mv^{2}+ \frac{1}{2}I\omega ^{2} = mgh$$\frac{1}{2}mv^{2}+ \frac{1}{2}m v^{2}k^{2}= mgh$that total K.E = mghalso $v= \sqrt{\frac{2mgh}{1+k^{2}}}$

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